Remark 6 (Connection with the
variational mapping particle filter) 
The variational mapping particle filter
[1]}
uses the nonparametric variational inference algorithm
proposed by [2]} to
generate approximate samples from a
non-Gaussian approximation of the
filtering distribution.
The idea is to minimize precisely
(REF )—a non-convex functional—by
implementing
a form of functional gradient descent on a reproducing kernel Hilbert space (RKHS) of transformations (with no restrictions to triangular maps).
The non-Gaussian approximation of the
unnormalized
filtering distribution is defined by the product of
the likelihood function with an approximation of
the forecast distribution at time \(k\)  given byFor ease of exposition,
we assume that the dynamic of the latent field is only
given at the observation times.
\(\pi _{Z_{k+1} \vert y_{1:k}}\approx \frac{1}{M}\,\sum _{i=1}^M\pi _{Z_{k+1} \vert Z_{k}}(\cdot \vert z^i),\) 
